论文标题
线性差分 - 代数系统通常可控
Linear differential-algebraic systems are generically controllable
论文作者
论文摘要
在目前的工作中，我们调查了形式的一组可控微分方algebraic系统的拓扑属性$ b \ in \ mathbb {r}^{\ ell \ times m} $。我们考虑五个可控概念的无初始定义（无穷大的可控性），冲动可控性，行为意义上的可控性，完全可控性和强大的可控性。为了能够利用这些概念的已知代数特征，我们首先考虑块矩阵，其条目是一种不确定的矩阵。我们发现，这种块矩阵的集合（其等级在有理函数甚至在整个复杂平面上都是“完整”的必要条件。使用这些结果，我们可以为上述五个可控性概念中的每个概念中的每个概念，分别在$ \ ell，n $和$ m $的情况下找到必要和足够的条件，根据该条件，一组可控系统是通用的。
In the present work we investigate topological properties of the set of controllable differential-algebraic systems of the form $\tfrac{\text{d}}{\text{d}t}Ex = Ax+Bu$ with real matrices $E,A\in\mathbb{R}^{\ell\times n}$ and $B\in\mathbb{R}^{\ell\times m}$. We consider the five controllability concepts free initializability (controllability at infinity), impulse controllability, controllability in the behavioural sense, complete controllability and strong controllability. To be able to make use of the already known algebraic characterizations of these concepts, we first consider block matrices whose entries are real polynomials in one indeterminant. We find necessary and sufficient conditions under which the set of such block matrices, whose rank is "full" in the field of rational functions or even on the whole complex plane, is generic. Using these results, we can then for each of the five controllability concepts mentioned above find necessary and sufficient conditions at $\ell,n$ and $m$, respectively, under which the set of controllable systems is generic.
